GCSE Mathematics Non-Calculator Foundation Tier Study Guide 2022

General Examination Information and Mathematical Formulae

• Examination Details: WJEC CBAC Ltd GCSE Mathematics Component 1 (Non-Calculator) Foundation Tier, Friday 20 May 2022. Duration: 2 hours 15 minutes.

• Permitted Equipment: Ruler, protractor, and a pair of compasses. The use of a calculator is not permitted.

• Formula List - Area and Volume:   - Curved surface area of a cone = $\pi r l$   - Surface area of a sphere = $4 \pi r^2$   - Volume of a sphere = $\frac{4}{3} \pi r^3$   - Volume of a cone = $\frac{1}{3} \pi r^2 h$   - Note: $r$ is the radius, $l$ is the slant height of a cone, and $h$ is the perpendicular height.

• Kinematics Formulae (Constant Acceleration):   - $v = u + at$   - $s = ut + \frac{1}{2} at^2$   - $v^2 = u^2 + 2as$   - Note: $a$ is constant acceleration, $u$ is initial velocity, $v$ is final velocity, $s$ is displacement, and $t$ is time.

Basic Arithmetic and Number Operations

• Fundamental Calculations:   - Multiplication: $3 \times 400 = 1200$   - Division by Powers of 10: $600 \div 1000 = 0.6$   - Order of Operations (BIDMAS): $10 + 4 \times 3 = 10 + 12 = 22$   - Directed Numbers (Subtractions with Negatives): $6 - (-7) = 6 + 7 = 13$

• Value Comparisons and Sets:   - To compare values, convert them to a common format (e.g., decimals):     - $\frac{3}{4} = 0.75$     - $\frac{1}{2} = 0.50$     - $0.35$     - $0.315$     - $0.6$   - In this set, the smallest value is $0.315$.

• Multi-Step Arithmetic Expression:   - Calculation: $80 + (25\% \text{ of } 48) - \frac{2}{5} \text{ of } 45$   - Step 1: $25\% \text{ of } 48 = \frac{1}{4} \times 48 = 12$   - Step 2: $\frac{2}{5} \times 45 = 2 \times 9 = 18$   - Step 3: $80 + 12 - 18 = 74$

• Square Roots: Value of $\sqrt{49} = 7$.

• Conversions:   - Fraction to Percentage: $\frac{11}{25} = \frac{44}{100} = 44\%$   - Percentage to Decimal: $87\% = 0.87$

• Calculations with Fractions:   - Finding the value of $\frac{2}{3} \times \frac{6}{2}$. Process involves multiplying numerators ($2 \times 6 = 12$) and denominators ($3 \times 2 = 6$). Result: $\frac{12}{6} = 2$.

Probability and Combinatorics

• Probability Scales:   - Arrow positions on a scale from $0$ to $1$ represent likelihood (e.g., $1 = \text{certain}$, $0 = \text{impossible}$, $0.5 = \text{even chance}$).

• Fair Spinner Outcomes:   - Spinner values provided: $1, 2, 3, 5, 8, 13$.   - Probability of spinning a number greater than $13$: $0$ (impossible).   - Probability of spinning an even number: There are two even numbers ($2, 8$) out of six possible values. Probability = $\frac{2}{6} = \frac{1}{3}$.

• Listing Outcomes:   - Mission: Pick a team of one girl (Poppy, Ruby, Sally, Zoe) and one boy (Tariq, Will).   - List of possibilities ($G, B$): $(P, T), (P, W), (R, T), (R, W), (S, T), (S, W), (Z, T), (Z, W)$.   - Total options: $8$.   - Probability of picking (Sally, Tariq): $\frac{1}{8}$.

Coordinate Geometry and 2D Shapes

• Kite ABCD Construction:   - Grid properties: $1\text{cm}$ square grid.   - Coordinate identified for point B: (e.g., from visual diagram analysis).   - Symmetry: Kite has one line of symmetry. If $BD = 6\text{cm}$, point D must be placed symmetrically across the axis of symmetry from B.   - Measuring Length: Calculation of distance CD based on the grid placement.

• Parallelogram Properties (ABCD):   - Diagonal AC splits the parallelogram into two triangles: $\triangle ABC$ and $\triangle CDA$.   - Equality statements for parallelograms:     - Opposing sides are equal: $AB = DC$ and $AD = BC$.     - The diagonal AC is shared by both triangles.     - $\triangle ABC$ is congruent to $\triangle CDA$.

Linear Relations and Formula Modeling

• Kebab Cooking Formula:   - $\text{Time} = 2.5 \times \text{Number of kebabs} + 16$   - Application for 10 kebabs: $\text{Time} = 2.5 \times 10 + 16 = 25 + 16 = 41 \text{ minutes}$.   - Application for 26 minutes total time:     - $26 = 2.5N + 16$     - $10 = 2.5N$     - $N = 4 \text{ kebabs}$.

• Subject of the Formula:   - Original: $t = 5 + 3n$   - To make $n$ the subject:     - $t - 5 = 3n$     - $n = \frac{t - 5}{3}$

Financial Mathematics and Utilities

• Historical Price Comparison (Train Journey):   - 2019 Cost = $£300$.   - 1979 Cost = $8\%$ of $£300$.   - Calculation: $0.08 \times 300 = £24$.

• Railcard Savings:   - Adult Ticket: $£15$ normal price, $\frac{1}{3}$ off discount. Saving = $\frac{1}{3} \times 15 = £5$. Cost = $£10$.   - Child Ticket: $£8$ normal price, $60\%$ off discount. Saving = $0.60 \times 8 = £4.80$. Cost = $£3.20$.   - Total Saving = $£5 + £4.80 = £9.80$.

• Wage Calculations (Rosheen):   - Weekday: Worked $6.5$ hours at $£9$/hour. Earnings = $6.5 \times 9 = £58.50$.   - Weekend total: $£314$ including $£160$ tips. Earnings from hours = $£314 - £160 = £154$.   - Weekend hourly rate: Worked $14$ hours. Rate = $154 \div 14 = £11$/hour.

• Bill Estimation (Mr. Cooke):   - Offer: Buy any $4$ burgers and get the $2$ cheapest free.   - Order: Mega Burger ($£8.99$), Vegan Burger ($£7.25$), Chicken Burger ($£8.99$), Regular Burger ($£6.30$), Fish Pie ($£9.90$), Veg Lasagne ($£6.80$).   - The $4$ burgers: $£8.99, £7.25, £8.99, £6.30$.   - Cheapest $2$ burgers (free): Regular ($£6.30$) and Vegan ($£7.25$).   - Bill items: $£8.99$ (Mega) + $£8.99$ (Chicken) + $£9.90$ (Fish Pie) + $£6.80$ (Lasagne) = $£34.68$. Rounded to nearest pound = $£35$.

• Utility Costs (Electricity Bill for Mo):   - Period: $30$ days.   - Standing Charge: $30 \text{ days} \times 20\text{p/day} = 600\text{p} = £6.00$.   - Usage Charge: $500 \text{ kWh} \times 14\text{p/kWh} = 7000\text{p} = £70.00$.   - Subtotal = $£76.00$.   - VAT ($5\%$): $0.05 \times 76 = £3.80$.   - Total Bill = $£76 + £3.80 = £79.80$.

• Exchange Rates and Comparisons:   - Rates: $€1 = £0.90$ and $\$1.25 = £1$.   - Silver Pencil Germany: $€110 \times 0.90 = £99.00$.   - Silver Pencil USA: $\$125 \div 1.25 = £100.00$.   - Conclusion: The pencil is cheaper in Germany.

Statistics and Data Handling

• Ages of the Cooke Family:   - Conditions: $5$ children total. Two children have the same age. Youngest is $12$. Range is $5$ years ($\max - \min = 5$). Mode is $14$.   - Range Calculation: $\text{Oldest} = 12 + 5 = 17$.   - Ages set: $(12, 14, 14, x, 17)$ where $x$ is a unique number between $12$ and $17$, excluding $14$.

• Data Criticisms: Identifying errors in sprint time graphs (e.g., overlapping intervals, missing labels, or inappropriate scales).

• Pie Chart Construction:   - Data for 30 students: Beach sports (12 students), Funfair rides (8 students), Water park (6 students), Cycling (4 students).   - Calculation of Angles: Total students = $30$. Total degrees = $360^{\circ}$. Angle per student = $360 \div 30 = 12^{\circ}$.   - Beach (B): $12 \times 12 = 144^{\circ}$   - Funfair (F): $8 \times 12 = 96^{\circ}$   - Water (W): $6 \times 12 = 72^{\circ}$   - Cycling (C): $4 \times 12 = 48^{\circ}$

Ratios, Proportions, and Fractions

• Investment Ratios:   - Theo invests $£45,000$, Jenny invests $£35,000$.   - Simplest Form: $45,000 : 35,000 = 45 : 35 = 9 : 7$.

• Profit Sharing:   - Jenny's profit ($7$ parts) = $£21,000$.   - Value of 1 part = $21,000 \div 7 = £3,000$.   - Theo's profit ($9$ parts) = $9 \times 3,000 = £27,000$.   - Difference in profit = $£27,000 - £21,000 = £6,000$.

• Loss Calculation:   - Jenny loses her first-year profit ($£21,000$) plus $\frac{3}{10}$ of original investment.   - Investment Loss: $\frac{3}{10} \times 35,000 = 10,500$.   - Total Loss = $21,000 + 10,500 = £31,500$.

• Value for Money Comparison:   - Conversion: $1 \text{ pint} = 600\text{ml}$. Thus, a half-pint glass = $300\text{ml}$.   - Glass: $300\text{ml}$ for $£1.50 \rightarrow 200\text{ml}$ for $£1.00$.   - Bottle: $500\text{ml}$ for $£2.00 \rightarrow 250\text{ml}$ for $£1.00$.   - Conclusion: The bottle offers more product per pound, making it better value.

• Workday Fraction Problems:   - Brad's time: $\frac{3}{7}$ designing, $\frac{5}{14}$ digging. Plants = Remaining time.   - Sum: $\frac{6}{14} + \frac{5}{14} = \frac{11}{14}$.   - Buying plants: $1 - \frac{11}{14} = \frac{3}{14}$.   - Aroon's time: Paperwork is $324$ minutes, representing $\frac{3}{5}$ of his day.   - Total Day (minutes): $\frac{324}{3} \times 5 = 108 \times 5 = 540 \text{ minutes}$.   - Total Day (hours): $540 \div 60 = 9 \text{ hours}$.

• Inverse Proportion (Machines and Work):   - 6 machines make 3000 erasers in 2 hours.   - 1 machine makes 3000 erasers in 12 hours.   - 1 machine makes 1000 erasers in 4 hours.   - 8 machines make 1000 erasers in $4 \div 8 = 0.5$ hours (30 minutes).

Geometry and Measurements

• Bearings calculations:   - Ship journey P to Q: bearing $100^{\circ}$, distance $8\text{km}$.   - Journey Q to R: bearing $160^{\circ}$, distance $8\text{km}$.   - Alternate angles: Angle $x$ is explained via parallel lines representing North. If the bearing P to Q is $100^{\circ}$, then the interior angle at Q from the North line southwards is derived via alternate angles to the North-segment.

• Cones and Scaled Models:   - Dimensions: Base radius $15\text{cm}$, Height $30\text{cm}$.   - Volume = $\frac{1}{3} \pi (15)^2 (30) = \frac{1}{3} \pi (225)(30) = 2250\pi \text{ cm}^3$.   - Scale drawing ratio $5 : 1$:     - Scale radius = $15 \div 5 = 3\text{cm}$.     - Scale height = $30 \div 5 = 6\text{cm}$.

• Fluid Dynamics and Graphs:   - Tank capacity = $225$ litres. Flow rate = $5$ litres/minute.   - Total time to empty = $225 \div 5 = 45 \text{ minutes}$.   - Time to decrease by $40\%$ of original: $0.40 \times 225 = 90 \text{ litres}$.   - Time required = $90 \div 5 = 18 \text{ minutes}$.

Advanced Algebra and Population Estimation

• Expansion and Simplification:   - $(4x + 5)(2x - 1) = 8x^2 - 4x + 10x - 5 = 8x^2 + 6x - 5$.

• Factorisation and Solving:   - Factorise $x^2 - 10x + 21$: Find factors of $21$ that sum to $-10$. Factors are $-3$ and $-7$.   - Result: $(x - 3)(x - 7)$.   - Equation solutions: $x = 3$ or $x = 7$.

• Capture-Recapture Population Estimate:   - Method: $\frac{\text{Marked in 1st sample (M)}}{\text{Total Population (N)}} = \frac{\text{Marked in 2nd sample (m)}}{\text{Total 2nd sample size (n)}}$.   - Data: $M = 12, n = 30, m = 9$.   - Equation: $\frac{12}{N} = \frac{9}{30}$.   - Calculation: $9N = 12 \times 30 = 360 \rightarrow N = 40$.   - Reliability Comment: Small sample sizes, the assumption of a closed population, or the possibility of marking impacting behavior can affect reliability.